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Published online by Cambridge University Press: 10 February 2020
While random matrices give us the exact value of the constant C(n), it is natural to search for alternate deterministic constructions that show that C(n)<n. This chapter explores this direction. The central notion here is that of spectral gap. To prove the key estimate that C(n)<n, it suffices to produce sequences of n-tuples of unitary matrices exhibiting a certain kind of spectral gap. The notion of quantum expanders naturally enter the discussion here. Their existence can be derived from that of groups with property (T) admitting sufficiently many finite dimensional unitary representations. The notion of quantum spherical code that we introduce hereis a natural way to describe what is needed in the present context.
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